Chaining Multiplications in Finite Fields with Chudnovsky-type Algorithms and Tensor Rank of the k-multiplication

TL;DR

This paper introduces Chudnovsky-type algorithms for multiplying multiple elements in finite fields, providing linear-rank tensor decompositions that improve the efficiency of finite field multiplication.

Contribution

It generalizes Chudnovsky algorithms to k-multiplcation, establishing uniform upper bounds and utilizing algebraic curves and function field towers for efficient computation.

Findings

Tensor rank of k-multiplcation is linear in n

Provides uniform upper bounds for finite field multiplication

Uses algebraic curves and function field towers for algorithm design

Abstract

We design a class of Chudnovsky-type algorithms multiplying k elements of a finite extension of order n a finite field K. We prove that these algorithms give a tensor decomposition of the k-multiplication for which the rank is linear in n uniformly in $q$. We give uniform upper bounds of the rank of k-multiplication in finite fields. They use interpolation on algebraic curves which transforms the problem in computing the Hadamard product of $k$ vectors with components in K. This generalization of the widely studied case of $k=2$ is based on a modification of the Riemann-Roch spaces involved and the use of towers of function fields having a lot of places of high degree.